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Ch 37: Special Relativity
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Young & Freedman Calc 15th EditionCh 37: Special Relativity문제 36
36장, 문제 36

Electrons are accelerated through a potential difference of 750750 kV, so that their kinetic energy is 7.50×1057.50\(\times\)10^5 eV.
(a) What is the ratio of the speed vv of an electron having this energy to the speed of light, cc?
(b) What would the speed be if it were computed from the principles of classical mechanics?

검증된 단계별 안내
1
Step 1: Understand the relationship between kinetic energy and speed for an electron. The kinetic energy (KE) of an electron accelerated through a potential difference is given by the equation KE = eV, where e is the charge of the electron and V is the potential difference.
Step 2: For part (a), use the relativistic energy-momentum relation to find the speed of the electron. The relativistic kinetic energy is given by KE = γmc² - mc², where γ is the Lorentz factor, m is the rest mass of the electron, and c is the speed of light. Solve for γ using the given kinetic energy.
Step 3: Calculate the Lorentz factor γ using the equation γ = 1 / sqrt(1 - (v²/c²)). Rearrange this equation to solve for the speed v of the electron in terms of c.
Step 4: For part (b), use classical mechanics to find the speed of the electron. The classical kinetic energy is given by KE = 0.5mv². Rearrange this equation to solve for v using the given kinetic energy.
Step 5: Compare the results from the relativistic and classical calculations to understand the difference in speeds. Note that relativistic effects become significant at speeds close to the speed of light, which is why the classical calculation may differ significantly from the relativistic one.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
11m
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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Potential Difference and Kinetic Energy

Potential difference, measured in volts, is the work done per unit charge to move a charge between two points. When electrons are accelerated through a potential difference, they gain kinetic energy equal to the charge multiplied by the potential difference. In this case, the kinetic energy of the electrons is given as 7.50 * 10^5 eV, which is derived from the potential difference of 750 kV.
추천 영상:
가이드 코스
07:24
Potential Energy Graphs

Relativistic Speed and Speed of Light

The speed of light, denoted as c, is approximately 3.00 * 10^8 m/s and represents the maximum speed at which information or matter can travel. When particles like electrons are accelerated to high energies, their speeds approach c, requiring relativistic physics to accurately describe their motion. The ratio of the electron's speed to c helps determine how relativistic effects influence the electron's behavior.
추천 영상:
가이드 코스
07:59
Speed Distribution & Special Speeds of Ideal Gases

Classical Mechanics vs. Relativistic Mechanics

Classical mechanics, based on Newton's laws, assumes that speeds are much less than the speed of light, allowing for straightforward calculations of velocity using kinetic energy. However, at high speeds, relativistic mechanics must be used, as it accounts for the increase in mass and energy effects at velocities approaching c. The question contrasts these two approaches to highlight the differences in calculated speeds.
추천 영상:
가이드 코스
06:24
Conservation Of Mechanical Energy
관련 실천
교과서 질문

A particle has rest mass 6.64×10−276.64\(\times\)10^{-27} kg and momentum 2.10×10−182.10\(\times\)10^{-18} kgm/s.

(a) What is the total energy (kinetic plus rest energy) of the particle?

(b) What is the kinetic energy of the particle?

(c) What is the ratio of the kinetic energy to the rest energy of the particle?

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교과서 질문

A proton (rest mass 1.67×10−271.67\(\times\)10^{-27} kg) has total energy that is 4.004.00 times its rest energy. What are (a) the kinetic energy of the proton; (b) the magnitude of the momentum of the proton; (c) the speed of the proton?

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교과서 질문

Compute the kinetic energy of a proton (mass 1.67×10−271.67\(\times\)10^{-27} kg) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) 8.00×1078.00\(\times\)10^7 m/s and (b) 2.85×1082.85\(\times\)10^8 m/s.

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교과서 질문

A force is applied to a particle along its direction of motion. At what speed is the magnitude of force required to produce a given acceleration twice as great as the force required to produce the same acceleration when the particle is at rest? Express your answer in terms of the speed of light.

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교과서 질문

An observer in frame S′ is moving to the right (+x-direction) at speed u = 0.600c away from a stationary observer in frame S. The observer in S′ measures the speed v′ of a particle moving to the right away from her. What speed v does the observer in S measure for the particle if (a) v′ = 0.400c; (b) v′ = 0.900c; (c) v′ = 0.990c?

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